The concept of emerging behaviours is often quoted within the literature of complex systems, and used as an attribute to assist with the description of these systems. The definition of what constitute an emerging behaviour has been debated in the literature, and we believe that it is, perhaps, the lack of mathematical definition of behaviour in the literature of complex systems what may be contributing to the confusion concerning the concept and even its need.
Hence, in this paper, we seek to point the readership to the literature of mathematical system theory that provides a behavioural perspective on systems and revisit the key mathematical definitions of behaviour. Such work uses set-theoretic concepts to define system behaviours as restrictions on, or subsets of, a universum set of variables that are in correspondence with quantitative attributes of a system. Such behaviours can relate not only to global attributes, but also include latent variables associated with hidden attributes.
We discuss how behaviours defined in this way are simply a consequence of a combination of components and structure for systems in general, and that the emerging behaviours, as described in the literature of complex systems, can indeed occur in systems classified as non-complex. We show this through simple examples of physical dynamical systems. We then discuss that the intractability of the properties of these behaviours due to system dimensionality is what could perhaps be a better qualifying attribute of complex systems.